Chapter one MAGNETIC CIRCUITS Dr. Gamal Mahmoud Ali Sowilam Electrical Engineering Department Contents 1. I-H RELATION 2. B-H RELATION 3. MAGNETIC EQUIVALENT CIRCUIT * Analogy between magnetic circuit and electric circuit. 4. MAGNETIZATION CURVE 5. MAGNETIC CIRCUIT WITH AIR GAP 6. INDUCTANCE 7. HYSTERESIS 8. SINUSOIDAL EXCITATION 9. PERMANENT MAGNET MAGNETIC CIRCUITS Rotating electrical machines, such as dc machines, induction machines, and synchronous machines, are the most important ones used to perform this energy conversion. The transformer, although not an electromechanical converter, plays an important role in the conversion process. Other devices, such as actuators, solenoids, and relays, are concerned with linear motion. In all these devices, magnetic materials are used to shape and direct the magnetic fields that act as a medium in the energy conversion process. A major advantage of using magnetic material in electrical machines is the fact that high flux density can be obtained in the machine, which results in large torque or large machine output per unit machine volume. In other words, the size of the machine is greatly reduced by the use of magnetic materials. in this chapter properties of magnetic materials are discussed and some methods for analyzing the magnetic circuits are outlined. In electrical machines, the magnetic circuits may be formed by ferromagnetic materials only (as in transformers) or by ferromagnetic materials in conjunction with an air medium (as in rotating machines). In most electrical machines, except permanent magnet machines, the magnetic field (or flux) is produced by passing an electrical current through coils wound on ferromagnetic materials. 1. i-H RELATION When a conductor carries current a magnetic field is produced around it. The direction of flux lines or magnetic field intensity H can be determined by what is known as the thumb rule. Thumb rule states that if the conductor is held with the right hand with the thumb indicating the direction of current in the conductor, then the fingertips will indicate the direction of magnetic field intensity The relationship between current and field intensity can be obtained by using Ampere's circuit law, which states that the line integral of the magnetic field intensity H around a closed path is equal to the total current linked by the contour. where H is the magnetic field intensity at a point on the contour and dl is the incremental length at that point. If θ is the angle between vectors Hand dI, then consider a conductor carrying current i as shown in Figure. To obtain an expression for the magnetic field intensity H at a distance r from the conductor, draw a circle of radius r, At each point on this circular contour, H and dl are in the same direction, that is, θ = 0. Because of symmetry, H will be the same at all points on this contour. Therefore: 2. B-H RELATION The magnetic field intensity H produces a magnetic flux density B everywhere it exists. These quantities are functionally related by: For free space or electrical conductors (such as aluminum or copper) or insulators, the value of µr is unity. for ferromagnetic materials (such as iron, cobalt, and nickel), the value of µr varies from several hundred to several thousand. For materials used in electrical machines, µr varies in the range of 2000 to 6000. A large value of µr implies that a small current can produce a large flux density in the machine. 3. MAGNETIC EQUIVALENT CIRCUIT a simple magnetic circuit having a ringshaped magnetic core, called a toroid, and a coil that extends around the entire circumference. When current i flows through the coil of N turns, magnetic flux is mostly confined in the core material. The flux outside the toroid, called leakage flux, is so small that for all practical purposes it can be neglected. Consider a path at a radius r. The magnetic intensity on this path is H and, from Ampere's circuit law, The quantity Ni is called the magnetomotive force (mmf) F, and its unit is ampere-turn. If we assume that all the fluxes are confined in the toroid, that is, there is no magnetic leakage, the flux crossing the cross section of the toroid is: where B is the average flux density in the core and A is the area of cross section of the toroid. The average flux density may correspond to the path at the mean radius of the toroid. where is called the reluctance of the magnetic path and P is called the permeance of the magnetic path. Analogy between magnetic circuit and electric circuit. electric circuit magnetic circuit 4. MAGNETIZATION CURVE If the magnetic intensity in the core is increased by increasing current, the flux density in the core changes in the way shown in Figure. B-H characteristic (magnetization curve). The flux density B increases almost linearly in the region of low values of the magnetic intensity H. However, at higher values of H, the change of B is nonlinear. The magnetic material shows the effect of saturation. The B-H curve, shown in Figure, is called the magnetization curve. The reluctance of the magnetic path is dependent on the flux density. It is low when B is low, high when B is high. The magnetic circuit differs from the electric circuit in this respect; resistance is normally independent of current in an electric circuit, whereas reluctance depends on the flux density in the magnetic circuit. The B-H characteristics of three different types of magnetic corescast iron, cast steel, and silicon sheet steel-are shown in Figure. Note that to establish a certain level of flux density B* in the various magnetic materials the values of current required are different. 5. MAGNETIC CIRCUIT WITH AIR GAP In electric machines, the rotor is physically isolated from the stator by the air gap. A cross-sectional view of a dc machine is shown in Figure. Practically the same flux is present in the poles (made of magnetic core) and the air gap. To maintain the same flux density, the air gap will require much more mmf than the core. If the flux density is high, the core portion of the magnetic circuit may exhibit a saturation effect. However, the air gap remains unsaturated, since the B-H curve for the air medium is linear µ is constant). A magnetic circuit having two or more media-such as the magnetic core and air gap in Figure is known as a composite structure. For the purpose of analysis, a magnetic equivalent circuit can be derived for the composite structure. Let us consider the simple composite structure of following Figure. The driving force in this magnetic circuit is the mmf, F = Ni, and the core medium and the air gap medium can be represented by their corresponding reluctances. The equivalent magnetic circuit is shown in Figure. Composite structure. Magnetic core with air gap. Magnetic equivalent circuit. In the air gap the magnetic flux lines bulge outward somewhat, as shown in Figure, this is known as fringing of the flux. The effect of the fringing is to increase the cross-sectional area of the air gap. Fringing flux. For small air gaps the fringing effect can be neglected. If the fringing effect is neglected, the cross-sectional areas of the core and the air gap are the same and therefore: EXAMPLE 1 The following figure represents the magnetic circuit of a primitive relay. The coil has 500 turns and the mean core path is lc = 360 mm. When the air gap lengths are 1.5 mm each, a flux density of 0.8 tesla is required to actuate the relay. The core is cast steel. (a) Find current in the coil. (b) Compute the values of permeability and relative permeability of the core. (c) If the air gap is zero, find the current in the coil for the same flux density (0.8 T) in the core. Solution The air gap is small and so fringing can be neglected. Hence the flux density is the same in both air gap and core. From the B-H curve of the cast steel core (Fig. 1.7). Note that although the air gap is very small compared to the length of the core (lg = 1.5 mm, Ic = 360 mm), most of the mmf is used at the air gap. (b) Permeability of core: Relative permeability of core: Note that if the air gap is not present, a much smaller current is required to establish the same flux density in the magnetic circuit. EXAMPLE 1.3 In the magnetic circuit of following figure, the relative permeability of the ferromagnetic material is 1200. Neglect magnetic leakage and fringing. All dimensions are in centimeters, and the magnetic material has a square cross sectional area. Determine the air gap flux, the air gap flux density, and the magnetic field intensity in the air gap. Solution The mean magnetic paths of the fluxes are shown by dashed lines in previous figure. The equivalent magnetic circuit is shown in following figure. From symmetry The loop equations are: OR 6. INDUCTANCE A coil wound on a magnetic core, such as that shown in Figure is frequently used in electric circuits. This coil may be represented by an ideal circuit element, called inductance, which is defined as the flux linkage of the coil per ampere of its current. Coil-core assembly. Equivalent inductance. defines inductance in terms of physical dimensions, such as cross-sectional area and length of core. defines inductance in terms of the reluctance of the magnetic path Note that inductance varies as the square of the number of turns. EXAMPLE For the magnetic circuit of Figure, N = 400 turns. Mean core length lg = 50 cm. Air gap length lg = 1.0 mm Cross-sectional area Ac = Ag = 15 cm^2. Relative permeability of core µr= 3000. i = 1.0 A Find (a) Flux and flux density in the air gap. (b) Inductance of the coil. Solution EXAMPLE The coil in Figure has 250 turns and is wound on a silicon sheet steel. The inner and outer radii are 20 and 25 cm, respectively, and the toroidal core has a circular cross section. For a coil current of 2.5 A. Find (a) The magnetic flux density at the mean radius of the toroid. (b) The inductance of the coil, assuming that the flux density within the core is uniform and equal to that at the mean radius. Solution Inductance can also be calculated using the other equation 7. HYSTERESIS (a) Core-coil assembly and exciting circuit. (b) Hysteresis. (c) Hysteresis loops. Magnetization and hysteresis loop * Assume that the core is initially un-magnetized. 1. If the magnetic intensity H is now increased by slowly, increasing the current i, the flux density will change according to the curve Oa in Figure. The point a corresponds to a particular value of the magnetic intensity, say H1 (corresponding current is i 1) . 2. If the magnetic intensity is now slowly decreased, the B-H curve will follow a different path, such as abc in Figure b. 3. When H is made zero, the core has retained flux density Br, known as the residual flux density. 4. If H is now reversed (by reversing the current i) the flux in the core will decrease and for a particular value of H, such as H; in Figure b, the residual flux will be removed. This value of the magnetic field intensity ( - Hc) is known as the coercivity or coercive force of the magnetic core. 5. If H is further increased in the reverse direction, the flux density will increase in the reverse direction. For current –i1 the flux density will correspond to the point e. 6. If H is now decreased to zero and then increased to the value H1, the B-H curve will follow the path efga’. The loop does not close. 7. If H is now varied for another cycle, the final operating point is a". The operating points a‘ and a" are closer together than points a and a'. 8. After a few cycles of magnetization, the loop almost closes, and it is called the hysteresis loop. The loop shows that the relationship between B and H is nonlinear and multivalued. Note that at point c the iron is magnetized, although the current in the coil is made zero. Throughout the whole cycle of magnetization, the flux density lags behind the magnetic intensity. This lagging phenomenon in the magnetic core is called hysteresis. Smaller hysteresis loops are obtained by decreasing the amplitude of variation of the magnetic intensity. A family of hysteresis loops is shown in Figure c. The locus of the tip of the hysteresis loop, shown dashed in Figure c, is called the magnetization curve. If the iron is magnetized from an initial unmagnetized condition, the flux density will follow the magnetization curve. In some magnetic cores, the hysteresis loop is very narrow. If the hysteresis effect is neglected for such cores, the B-H characteristic is represented by the magnetization curve. Deltamax Cores Figure d: B-H loop for a deltamax core (50% Fe and 50% Ni). Special ferromagnetic alloys are sometimes developed for special applications. The hysteresis loops for these alloys have shapes that are significantly different from those shown in Figure b and c. An alloy consisting of 50% iron and 50% nickel has the B-H loop shown in Figure d Cores made of alloys having this type of almost square B-H loop are known as deltamax cores. A coil wound on a deltamax core can be used as a switch. Note that when the flux density is less than the residual flux density (B < Br ) the magnetic intensity (and hence the current) is quite low. As the flux density exceeds the residual flux density (B > Br ), the magnetic intensity (hence the current) increases sharply. This property can be exploited to make a coil wound on a deltamax core behave as a switch (very low current when the core is unsaturated and very high current when the core is saturated). 8. SINUSOIDAL EXCITATION In ac electric machines as well as many other applications, the voltages and fluxes vary sinusoidally with time. From Faraday's law, the voltage induced in the N-turn coil is Note that if the flux changes sinusoidally, the induced voltage changes cosinusoidally. The root mean square (rms) value of the induced voltage is: This is an important equation and will be referred to frequently in the theory of ac machines. EXAMPLE A single phase 120 V, 60 Hz supply is connected to the coil of Figure. The coil has 200 turns. The parameters of the core are as follows: Length of core = 100 cm Cross-sectional area of core = 20 cm*cm Relative permeability of core = 2500 (a) Obtain an expression for the flux density in the core. (b) Obtain an expression for the current in the coil. Solution (a) From the Erms= 4.44Nfmax max =Erms/4.44Nf= 120/(4.44*200*60) = 0.002253 Wb (b) = 358.575 At/m